Partial correlation based penalty functions and prior distributions for Gaussian graphical models

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Abstract

Graphical models are a useful tool for encoding conditional independence relations. A common goal is to select the graphical model that best describes the conditional independence relationships between variables given observations of these variables. Under the additional Gaussian assumption, conditional independence is equivalent to zero entries in the inverse covariance matrix Ɵ. Thus sparse estimation of Ɵ in turn specifies a graphical model and the associated conditional independencies. Popular frequentist methods for this often involve placing a penalty function on Ɵ and maximising a penalised likelihood, whilst Bayesian methods require specification
of a prior distribution on Ɵ.

Conditional independence relations are invariant to non-zero scalar multiplication of the variables, however in this thesis we show that essentially all current penalised likelihood methods and many prior distributions are not invariant to such transformations of the variables. In fact many methods are very sensitive to rescaling of the variables which can, and often does, result in a vastly different selected graphical model. To remedy this issue we introduce new classes of penalty functions and prior distributions which are based on partial correlations. We show that such penalty functions and prior distributions lead to scale invariant estimation and posterior inference on Ɵ.

We pay particular attention to two penalty functions in this class. The partial correlation graphical LASSO places an L1 penalty on the partial correlations whilst the spike and slab partial correlation graphical LASSO is a penalty function based on a spike and slab prior formulation. The performance of these penalty functions is compared to that of current popular penalty functions in simulated and real world settings. We also investigate spike and slab priors in general for Gaussian graphical models and point out that care must be taken when considering the positive definiteness of Ɵ. With this in mind we provide some theoretical results based on Wigner matrices.

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